Answer
${\frac{dP}{dt}}={2RI{\frac{dI}{dt}}+I^2{\frac{dR}{dt}}}$
${{\frac{dR}{dt}}=-\frac{2P}{I^3}{\frac{dI}{dt}}}=-\frac{2R}{I}{\frac{dI}{dt}}$
Work Step by Step
We know that $P=I^2R$
on differentiating each side with respect to time:
${\frac{dP}{dt}={R{\frac{dI^2}{dt}}+I^2{\frac{dR}{dt}}}}$
${\frac{dP}{dt}}={2RI{\frac{dI}{dt}}+I^2{\frac{dR}{dt}}}$
when P is constant, its derivative is zero, so the above equation becomes:
$0={2RI{\frac{dI}{dt}}+I^2{\frac{dR}{dt}}}$
${2RI{\frac{dI}{dt}}=-I^2{\frac{dR}{dt}}}$
${{\frac{dR}{dt}}=-\frac{2R}{I}{\frac{dI}{dt}}}$
${{\frac{dR}{dt}}=-\frac{2RI}{I^2}{\frac{dI}{dt}}}$
since $\frac{P}{I}= IR$
${{\frac{dR}{dt}}=-\frac{2P}{I^3}{\frac{dI}{dt}}}$
The final answer:
${\frac{dP}{dt}}={2RI{\frac{dI}{dt}}+I^2{\frac{dR}{dt}}}$
${{\frac{dR}{dt}}=-\frac{2P}{I^3}{\frac{dI}{dt}}}=-\frac{2R}{I}{\frac{dI}{dt}}$
